Absolutely convergent Fourier series. An improvement of the Beurling--Helson theorem

Details Absolutely convergent Fourier series. An improvement of the Beurling--Helson theorem Absolutely convergent Fourier series. An improvement of the Beurling--Helson theorem

2011-12-21

arxiv.org/abs/1112.4892v3

Mathematics

Classical Analysis and ODEs

A typo on page 14 line 4 is corrected. A typo on page 16 line 5 is corrected. Notation in the proof of Lemma 2 are replaced wi

Scientific paper

We consider the space $A(\mathbb T)$ of all continuous functions $f$ on the circle $\mathbb T$ such that the sequence of Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z\}$ belongs to $l^1(\mathbb Z)$. The norm on $A(\mathbb T)$ is defined by $\|f\|_{A(\mathbb T)}=\|\hat{f}\|_{l^1(\mathbb Z)}$. According to the known Beurling--Helson theorem, if $\phi : \mathbb T\rightarrow\mathbb T$ is a continuous mapping such that $\|e^{in\phi}\|_{A(\mathbb T)}=O(1), ~n\in\mathbb Z,$ then $\phi$ is linear. It was conjectured by Kahane that the same conclusion about $\phi$ is true under the assumption that $\|e^{in\phi}\|_{A(\mathbb T)}=o(\log |n|)$. We show that if $\|e^{in\phi}\|_{A(\mathbb T)}=o((\log\log |n|/\log\log\log |n|)^{1/12})$ then $\phi$ is linear.

Affiliated with

Also associated with

No associations

Rating

Absolutely convergent Fourier series. An improvement of the Beurling--Helson theorem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Absolutely convergent Fourier series. An improvement of the Beurling--Helson theorem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Absolutely convergent Fourier series. An improvement of the Beurling--Helson theorem will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-189801